A presentation for the mapping class group of the closed non-orientable surface of genus 4

Abstract

Communicated by E.M. Friedlander MSC: Primary: 57N05 secondary: 20F05 20F38 a b s t r a c t In [B. Szepietowski, A presentation for the mapping class group of a non-orientable surface from the action on the complex of curves, Osaka J. Math. 45 (2008) 283-326] we proposed a method of finding a finite presentation for the mapping class group of a non-orientable surface by using its action on the so called ordered complex of curves. In this paper we use this method to obtain an explicit finite presentation for the mapping class group of the closed non-orientable surface of genus 4. The set of generators in this presentation consists of 5 Dehn twists, 3 crosscap transpositions and one involution, and it can be immediately reduced to the generating set found by Chillingworth [D.R.J. Chillingworth, A finite set of generators for the homeotopy group of a non-orientable surface, Proc. Camb. Phil. Soc. 65 (1969) 409-430].

Automorphisms of the mapping class group of a nonorientable surface

“…where Z l is the subgroup generated by the boundary twists of S C (see [12,Section 4]). We also have a monomorphism Mod(S C )/Z l → Mod(S ′ ), where S ′ is the surface obtained from S C by collapsing each boundary component to a puncture.…”

Counting pseudo-Anosov mapping classes on the 3-punctured projective plane

“…Since every h ∈ Homeo(M, {p}) may be extended by the identity on the disc to h ′ ∈ Homeo(F, {p, q}), we have a homomorphism PM(M, {p}) → PM(F, {p, q}), which fits in the following short exact sequence (see [9,Section 7])…”

Generating the mapping class group of a nonorientable surface by crosscap transpositions

“…Because a crosscap transposition is the product of a crosscap slide and a Dehn twist, crosscap transpositions can be used instead of crosscap slides. The presentations of M(N g,n ) given in [13,18,19] use Dehn twists and crosscap transpositions as generators. Building on [13], Stukow [17] found a finite presentation of M(N g,0 ) and M(N g,1 ) with Dehn twists and one crosscap slide as generators.…”